Volkswagen AG is near resolving a criminal investigation into the German auto giant’s emissions cheating, said people familiar with the matter, aiming to reach an accord as soon as next week that would require the company to pay a financial penalty of several billion dollars and cap a major aspect of a long-running crisis.

Volkswagen, which has admitted to the environmental failures, is in ongoing discussions with the U.S. Justice Department to settle an expected criminal case and aiming to complete a deal before the Obama...